In this paper, we presented two new rectangular LB schemes based on raw moments and central moments, designated as RNR-LBM and RC-LBM, respectively, where the collision step involves relaxation of the corresponding moments, each with its own individual rates. Unlike the other existing schemes, they are constructed using a non-orthogonal moment basis and the moment equilibria are directly obtained via matching with those of the continuous Maxwell distribution function, thereby involving higher order velocity terms and without the use of many free parameters. By using a Chapman-Enskog analysis, correction terms are derived to eliminate the grid anisotropy introduced on the viscous stress tensor arising from the use of the rectangular lattice and the cubic non-Galilean invariant terms due to aliasing effects on the standard D2Q9 lattice. Such correction terms, which are incorporated via extending the equilibria for the second order moments, along with the expressions for the transport coefficients have simpler functional relationships involving the grid aspect ratio, the speed of sound and the diagonal components of the velocity gradient tensor. Formulas are derived to compute the latter locally based on non-equilibrium moments. Furthermore, the attendant transformation matrices that map between the distribution functions and the moments and parameterized by the grid aspect ratio are also considerably simplified owing to the use of non-orthogonal moment basis. All these considerations result in more robust and efficient implementations of the proposed RNR-LBM and RC-LBM when compared to the other existing rectangular LB formulations. These two schemes are validated against a variety of benchmark flow problems yielding accurate solutions for a wide range of grid aspect ratio. Furthermore, simulations demonstrate improvements in accuracy and significantly greater numerical stability regime for shear driven flows with the use of the RC-LBM when compared to RNR-LBM. Moreover, the effectiveness of using of our rectangular LB scheme in lieu of that based on the square lattice in reducing the computational cost is shown. The present rectangular central moment LB formulation can be extended to a 3D cuboid LB approach, which will be reported in the near future. The approach presented here involving the RNR-LBM and RC-LBM is for athermal flows. It can also be extended to include temperature variations either by using extended lattice sets or using dual distribution functions-based formulations in a rectangular lattice for efficient simulations of flow with heat transfer. Furthermore, while the present approach allows local variations in the viscosity via the relaxation parameters, a pressure-based rectangular central moment LB formulation involving variations in various fluid properties can be constructed using a modified kinetic equation and equilibria via a transformation similar to that presented for the square lattice in Ref. [1]}. Moreover, the development of multiblock rectangular/cuboid lattice based central moment LB schemes represents another interesting area. These are important topics for practical applications and are subjects for future investigations.
